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(in C kg
−
1
). We will see later that the flow of the pore
water relative to the grain framework drags an effec-
tive charge density
Q
0
As shown by Leroy and Revil (2004) and Leroy et al.
(2007), the previous set of equations can be solved
numerically using the parameters given in Table 1.2 as
input parameters. The parameters of Table 1.2 have been
optimized from a number of experimental data, espe-
cially zeta potential resulting from electrokinetic mea-
surements and surface conductivity data (see Leroy &
Revil, 2004; Leroy et al., 2007), and remain unchanged
in the present work. The output parameters of the
numerical TLM are the surface site densities in the Stern
and diffuse layers and therefore the partition coefficient
f
.
Some TLM computations of the fractions of counterions
in the Stern layer show that
f
is typically in the range
0.80
V
. We expect that for permeable
porous media, we have
Q
V
Q
V
1 72
In sandstones, the diffuse layer is relatively thin with
respect to the size of the pores and especially for the
pores that controlled the flow of the pore water. In
other words, most of the water is neutral with the
exception of the pore water surrounding the surface
of the grains. In addition, only a small fraction of the
diffuse layer is carried along the pore water flow. We
will see that the charge density
Q
V
should be under-
stood as an effective charge density that is controlled
by the flow properties (especially the permeability)
and that has little to do with the CEC itself. It should
be clear therefore that the CEC cannot be determined
from the effective charge density
Q
V
, which will be
properly defined later.
2
There is an excess of electrical conductivity in the
vicinity of the pore water
-
mineral interface responsible
for the so-called surface conductivity (see Figure 1.1).
This surface conductivity exists for any minerals in
contact with water including clean sands. That said,
the magnitude of surface conductivity is much stronger
in the presence of clay minerals due to their very high
surface area (surface area of the pore water
-
mineral
interface for a given pore volume).
3
The double layer is responsible for the (nondielectric)
low-frequency polarization of the porous material.
This polarization is coming from the polarization of
the electrical double layer in the presence of an electri-
cal field applied to the porous material. In the case of
seismoelectric effects, it implies a phase lag between
the pore fluid pressure and the electric field, but this
phase lag is expected to be small (typically <10 mrad
except at very low salinities where the magnitude of
the phase can reach 30 mrad) and most of the time
is neglected (for instance, by Pride, 1994).
The first consequence is fundamental to understanding
the nature of electrical currents associated with the flow
of pore water relative to the mineral framework (termed
streaming currents); therefore, the occurrence of electro-
kinetic (macroscopic) electrical fields is due to the flow of
pore water relative to the mineral framework associated
0.99, indicating that clay minerals have a much
larger fraction of counterions in the Stern layer by
comparison with glass beads at the same salinities. The
values of the partition coefficient determined from the
present model are also consistent with values determined
by other methods, for instance, using radioactive tracers
(Jougnot et al., 2009) and osmotic pressure (Gonçalvès
et al., 2007; Jougnot et al., 2009). This shows that the
present electrochemical model is consistent because it
can explain a wide diversity of properties.
-
1.1.3 Implications
As discussed in Sections 1.1.1 and 1.1.2, all minerals of
a porous material in contact with water are coated by
theelectricaldoublelayershowninFigure1.1.The
surface of the mineral is charged (due to isomorphic
substitutions in the crystalline network or surface
ionization of active sites such as hydroxyl >OH sites).
The surface charge is balanced by charges located in
the Stern layer and in the diffuse layer. There are three
fundamental implications associated with the existence
of this electrical double layer at the surface of silicates
and clays:
1
Pore water is never neutral. There is an excess of
charge in the pore water that can be written as
1
−
ϕ
ϕ
Q
V
=1
−
f
ρ
g
CEC
1 71
where
f
denotes the fraction of counterions in the
Stern layer (attached to the grains) and therefore
(1
f
) denotes the fraction of charge contained in
the diffuse layer,
-
ρ
g
denotes the mass density of the
grains (kg m
−
3
),
denotes the porosity, and CEC
denotes the cation exchange capacity of the material
ϕ
